Let’s analyze the problem step by step. The task is to study the convergence of the given integrals:

Part A:
$\int_0^1 \frac{\ln(x)}{x^{4/5}} \, dx$

Part B:
$\int_1^\infty \frac{\ln(x)}{x^{4/5}} \, dx$

Analysis for Part A:

1. Behavior near $x = 0$:
   The function $\frac{\ln(x)}{x^{4/5}}$ has a potential singularity at $x = 0$, as $\ln(x) \to -\infty$ and $x^{-4/5} \to \infty$.
   To study convergence, expand the integral: $\int_0^1 \frac{\ln(x)}{x^{4/5}} \, dx$

   Let $x = t^5$, so $dx = 5t^4 \, dt$. Under this substitution: $\ln(x) = \ln(t^5) = 5\ln(t)$ and $x^{4/5} = t^4$. Substituting: $\int_0^1 \frac{\ln(x)}{x^{4/5}} \, dx = \int_0^1 \frac{5\ln(t)}{t^4} \cdot 5t^4 \, dt = 25 \int_0^1 \ln(t) \, dt$

   This integral is divergent because $\ln(t) \to -\infty$ as $t \to 0$.

   Conclusion:
   The integral diverges.

Analysis for Part B:

1. Behavior as $x \to \infty$:
   The function $\frac{\ln(x)}{x^{4/5}}$ diminishes as $x \to \infty$, since $x^{4/5}$ dominates $\ln(x)$.
   To check convergence, examine: $\int_1^\infty \frac{\ln(x)}{x^{4/5}} \, dx$

   Let $x = e^t$, so $\ln(x) = t$ and $dx = e^t \, dt$. Substituting: $\int_1^\infty \frac{\ln(x)}{x^{4/5}} \, dx = \int_0^\infty \frac{t}{(e^t)^{4/5}} \cdot e^t \, dt = \int_0^\infty t e^{-t/5} \, dt$

   Let $u = t/5$, so $t = 5u$ and $dt = 5 \, du$. Substituting: $\int_0^\infty t e^{-t/5} \, dt = \int_0^\infty 5u e^{-u} \cdot 5 \, du = 25 \int_0^\infty u e^{-u} \, du$

   The integral $\int_0^\infty u e^{-u} \, du$ converges (it's a Gamma function, $\Gamma(2) = 1$).
   Hence: $\int_1^\infty \frac{\ln(x)}{x^{4/5}} \, dx = 25 \cdot 1 = 25$

   Conclusion:
   The integral converges, and its exact value is 25.

Final Answers:

• Part A: Divergent (input 333).
• Part B: Convergent, exact value is 25.

Would you like a more detailed breakdown of the calculations?

Related Questions:

1. What substitution techniques are helpful in integrals involving logarithms?
2. How can you determine divergence at a boundary point?
3. What is the Gamma function, and how is it used in definite integrals?
4. How does the rate of growth of logarithmic functions compare to polynomial or exponential functions?
5. Can the integral in Part A be regularized or reinterpreted to make it finite?

Tip: Always check for singularities at boundaries when analyzing improper integrals.